Free modules with two distinguished submodules

Abstract

Over a commutative ring R with identity, free modules M with 2 distinguished submodules are studied. The category Rep₂R of such objects M have the obvious morphisms between them, which are homomorphisms between .R-modules preserving the distinguished submodules. The endo-morphisms for each M constitute a subalgebra End_RM of the algebra End_RM and the readability of λ-generated R-algebras A as End_RM is considered for cardinals λ. Despite the fact that 4 is the minimal number of distinguished submodules for realizing any algebra over a field il, we are able to prove a similar result in Rep₂R for many rings R including R = Z and algebras which are cotorsion-free. Several examples illustrate the boarder line of our main result. The main theorem is applied for constructing Butler groups in [11]

Keywords:

• Modules with two distinguished submodules
• decomposition into i ndecomposables. Supported by a project No. G-0294-081.06/93 of the German-Israeli Foundation for Scientific Research & Development

1991 Mathematics Subject Classification.:

• Primary16G60
• 13C05
• 20K15
• 20K20
• Secondary 20K25
• 20K26
• 15A36